Evaluate: \(\dfrac{3.8^2 - 1.2^2}{3.8 - 1.2}\)
Answer: D
The numerator is a difference of two squares, so it factorises and the denominator cancels - no squaring is needed at all.
\[ \frac{a^{2}-b^{2}}{a-b} = \frac{(a+b)(a-b)}{a-b} = a+b \]
With \( a = 3.8 \) and \( b = 1.2 \):
\[ \frac{3.8^{2}-1.2^{2}}{3.8-1.2} = 3.8 + 1.2 = 5 \]
Checking the long way: \( 14.44 - 1.44 = 13 \) and \( 13 \div 2.6 = 5 \).
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What is difference between biggest and smallest fraction among \(\dfrac 23, \dfrac 34, \dfrac 45\) and \(\dfrac 56\) ?
Answer: C
\(\dfrac 23=0.66\\ \dfrac 34=0.75\\ \dfrac 45=0.8\\ \dfrac 56=0.833\)
So biggest is \(\dfrac 56\) and smallest is \(\dfrac 23\)
Their difference is \(\dfrac 56 - \dfrac 23 = \dfrac 16\)
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The expression (11.98 x 11.98 + 11.98 x x + 0.02 x 0.02) will be a perfect square for x equal to:
Answer: C
The expression is set up to be the square of a sum. Comparing with
\[ (a+b)^{2} = a^{2} + 2ab + b^{2} \]
the outer terms give \( a = 11.98 \) and \( b = 0.02 \), so the middle term must be
\[ 2ab = 2(11.98)(0.02) = 0.4792 \]
The middle term as written is \( 11.98x \), so
\[ 11.98x = 0.4792 \quad\Longrightarrow\quad x = 2(0.02) = 0.04 \]
The square is then \( (11.98 + 0.02)^{2} = 12^{2} = 144 \). Notice the shortcut: \( x \) is simply twice the smaller number, so the decimals never need multiplying out.
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The expression (11.98*11.98+11.98*x+0.02*0.02) will be a perfect square for x equal to:
Answer: A
The expression is set up to be the square of a sum. Comparing with
\[ (a+b)^{2} = a^{2} + 2ab + b^{2} \]
the outer terms give \( a = 11.98 \) and \( b = 0.02 \), so the middle term must be
\[ 2ab = 2(11.98)(0.02) = 0.4792 \]
The middle term as written is \( 11.98x \), so
\[ 11.98x = 0.4792 \quad\Longrightarrow\quad x = 2(0.02) = 0.04 \]
The square is then \( (11.98 + 0.02)^{2} = 12^{2} = 144 \). Notice the shortcut: \( x \) is simply twice the smaller number, so the decimals never need multiplying out.
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4211.01 + 22.261 - ? = 2645.759
Answer: A
The unknown is being subtracted, so it equals everything on the left minus the result on the right.
\[ 4211.01 + 22.261 - x = 2645.759 \]
\[ x = 4211.01 + 22.261 - 2645.759 \]
\[ 4211.010 + 22.261 = 4233.271 \]
\[ 4233.271 - 2645.759 = 1587.512 \]
The three near-identical distractors differ only in the last three digits, so the subtraction has to be carried out to full precision rather than estimated.
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Find the product: (i) 2.61 * 1.3(ii) 2.1693 * 1.4(iii) 0.4 * 0.04 * 0.004 * 40
Answer: C
Multiply as whole numbers, then place the decimal point using the total number of decimal places in the factors.
(i) \( 261 \times 13 = 3393 \); two places plus one is three, so
\[ 2.61 \times 1.3 = 3.393 \]
(ii) \( 21693 \times 14 = 303702 \); four places plus one is five, so
\[ 2.1693 \times 1.4 = 3.03702 \]
(iii) \( 4 \times 4 \times 4 \times 40 = 2560 \); one, two and three places with none from the 40 gives six, so
\[ 0.4 \times 0.04 \times 0.004 \times 40 = 0.002560 \]
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What value will come in place of question mark in the following equation?
(i) 0.006 ÷ ? = 0.6 (ii) ? ÷ 0.025 = 80
Answer: B
Rearrange each equation for the unknown.
(i) From \( 0.006 \div ? = 0.6 \):
\[ ? = \frac{0.006}{0.6} = 0.01 \]
(ii) From \( ? \div 0.025 = 80 \):
\[ ? = 80 \times 0.025 = 2 \]
The two rearrange in opposite directions: in the first the unknown is the divisor and so divides across, in the second it is the dividend and so multiplies.
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What value will replace the question mark in the following equations ?
(i) 5172.49 + 378.352 + ? = 9318.678 (ii) ? − 7328.96 = 5169.38
Answer: B
Isolate the question mark in each equation.
(i)
\[ ? = 9318.678 - 5172.49 - 378.352 = 3767.836 \]
(ii)
\[ ? = 5169.38 + 7328.96 = 12498.34 \]
Note the direction of each operation: the first needs subtraction because the unknown sits on the adding side, and the second needs addition because it is being subtracted from.
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If \(\dfrac {2994}{14.5} = 172\), then \(\dfrac {29.94}{1.45} = ?\)
Answer: C
\(\dfrac {29.94}{1.45}\\~\\= \dfrac {299.4}{14.5} \\~\\ =\left ( \dfrac {2994}{14.5} \times \dfrac {1}{10} \right )\\~\\=\dfrac {172}{10} \left [ \because \dfrac {2994}{14.5} = 172 \right ]\\~\\= 17.2\)
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\(4.\overline{86} - 3.\overline{71}\text{ = ?}\)
Answer: D
\(4.\overline{86} = \left(4 + \dfrac{86}{99}\right)\)
\(3.\overline{71} = \left(3 + \dfrac{71}{99}\right)\)
Hence,
\(4.\overline{86} - 3.\overline{71} \\~\\= \left(4 + \dfrac{86}{99}\right) - \left(3 + \dfrac{71}{99}\right)\\~\\ = \left( 1 + \dfrac{15}{99}\right)\\~\\ = 1.\overline{15}\)
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